This book introduces convex polytopes and their graphs, alongside the results and methodologies required to study them. It guides the reader from the basics to current research, presenting many open problems to facilitate the transition. The book includes results not previously found in other books, such as: the edge connectivity and linkedness of graphs of polytopes; the characterisation of their cycle space; the Minkowski decomposition of polytopes from the perspective of geometric graphs; Lei Xue's recent lower bound theorem on the number of faces of polytopes with a small number of vertices; and Gil Kalai's rigidity proof of the lower bound theorem for simplicial polytopes. This accessible introduction covers prerequisites from linear algebra, graph theory, and polytope theory. Each chapter concludes with exercises of varying difficulty, designed to help the reader engage with new concepts. These features make the book ideal for students and researchers new to the field.

‘This is a welcome monograph on combinatorial aspects of polytope theory, with a focus on graphs. While the book is written to be accessible to non-experts, it does capture a lot of the research on that subject from the last 30 years.’

Michael Joswig - Technische Universität Berlin and Max-Planck-Institute for Mathematics in the Sciences, Leipzig

‘Polytopes are geometric objects that have fascinated mathematicians since antiquity. Graphs are combinatorial objects that are ubiquitous throughout pure and applied mathematics. Ever since Steinistz's famous theorem asserting that planar 3-connected graphs are precisely graphs of three-dimensional polytopes, the match between polytopes and graphs was engraved in heaven.   Guillermo Pineda Villavicencio, a notable researcher exploring this connection, has authored a remarkable book on the subject. This book is well-suited for advanced undergraduates as well as professional mathematicians. It commences with a comprehensive introduction to polytopes and proceeds to explore the properties of polytopal graphs. Additionally, it delves into advanced topics, such as connectivity, diameter, reconstruction, decomposition, and more. I highly recommend this book.’

Gil Kalai - Hebrew University of Jerusalem and Reichman University